I am having a little problem with determining a trajectory.
I have a 2D curve, say $\{x(\lambda), y(\lambda)\}$ where $\lambda$ is not time; it is only a parameter that describes the evolution of the ...

I'm creating a program where I need to calculate the equation of the plane tangent to the earth at a given latitude and longitude. I used Projecting an Arbitrary Latitude and Longitude onto a Tangent ...

This shape, which I call the multiplicoid, is the equivalent of, and very similar to, an ellipse. However, instead of the distance between each point and the two focal points summing to a constant, ...

Could someone please give me the intuition behind using integral of squared second derivative as a measure of curve smoothness?
I was thinking that since curvature measures how fast a curve changes, ...

How to compute the length of a curve given by the formula
$$ f: (0, \frac{\pi}{2}) \ni t \rightarrow ( \cos^3t,\sin^3t) \in \Bbb R^2 $$
I know that the length of a curve in with image in $\Bbb R $ is ...

Under the conditions of Green’s Theorem, the area of a region $R$ enclosed by a
curve $C$ is
$$\oint_C x \, dy=-\oint_C y \, dx=\frac{1}{2}\oint_C (x \, dy - y \, dx)$$
I tried to use the result to ...

Given two separate annulus with centers $[C_1,C_2]$ and their corresponding radii being $[R_1,r_1]$ and $[R_2,r_2]$ respectively, larger radius being $R$. There are methods to look at whether they are ...

Is there a way to write out Laplace-Beltrami operator explicitly for a sufficiently smooth plane curve given by implicit equation $s(x,y)=0$?
I know that if we knew the parametrization of the curve, ...

Let $E_1$ be an ellipse fixed in the plane.
Let $E_2$ be a second, possibly different ellipse, which rolls around
without slippage
outside $E_1$, touching perimeter-to-perimeter. Let $c_2(t)$ be the ...

(Background: I am trying to understand the definition of angle-preserving function..I posted a question earlier but I still have doubts)
My question is:how is the angle between two curves defined if ...

Let $\mathbf{r}:[a,b]\to\mathbb{R}^2$ be a $C^2([a,b])$ regular curve. Is it true that $\mathbf{r}$ is convex if and only if its curvature $\kappa(t)\leq 0, \forall t\in [a,b]$ or $\kappa(t)\geq 0, ...

Let $\gamma :[a,b]\to\mathbb{R}^2$ be a $C^{1} ([a,b])$ injective application. Prove that there is another continuous parametrization $\rho:[c,d]\to\mathbb{R}^2$ such that the following two sets are ...